Physics 6730 Chaotic Pendulum

The damped, driven true pendulum is a simple mechanical system. Nonetheless, under appropriate conditions, it shows surprisingly complicated, chaotic motion.

The state of a true pendulum is described by the angle of suspension $\theta$, measured from the vertical and the corresponding velocity $\omega = d\theta/dt$. Thus $\theta = 0$ is the position of downward, stable equilibrium and $\theta = \pm\pi$ is the position of upward, unstable equilibrium.

With friction and a harmonic driving term the motion of the pendulum is described by the equation of motion

$$\frac{d^2 \theta}{dt^2} = -\sin \theta -\frac{d\theta}{q\, dt} + g \cos(\omega_d t).$$ (1)

If we introduce the angular velocity $\omega$ we can rewrite the second order equation as a pair of first order equations:

$$d\theta/dt = \omega$$ (2)

$$d\omega/dt = -\sin \theta -\omega/q + g \cos(\omega_d t)$$ (3)

On the rhs the gravitational acceleration is represented by $-\sin \theta$. By a judicious selection of time units, this term has been rendered dimensionless. The second term on the rhs introduces damping, parameterized by the dimensionless Q-factor $q$. If $q$ is small, damping is large. If it is large, damping is small. The third term on the rhs represents a harmonic driving term with strength $g$ (not to be confused with the gravitational acceleration) and angular frequency $\omega_d$.

We generally seek a solution of this equation $\theta(t)$ after specifying the initial state of the pendulum, say at $t = 0$. That makes this an initial value problem. As with all second order initial value problems we must specify two conditions, which are generally taken to be the initial values $\theta(0)$ and $\omega(0)$. Here is a sample choice for the parameters and initial values:

$\theta(0)$	0
$\omega(0)$	2
$g$	0.5
$q$	2
$\omega_d$	0.667

This choice leads to stable motion with a transient that gives way after a short while to steady state periodic motion.

The combination of the anharmonic character and the driving term allows for chaotic solutions. To see one, try increasing the strength of the driving term to $g = 1.5$. The solution should be extended to at least $t = 100$ to get a good feeling for its character.

Notice that the angle $\theta$ ranges over the entire real line, but the problem is really periodic. In plotting the results it may prove useful to map the solution onto the standard domain $[-\pi,\pi]$.

In addition to plotting $\theta$ as a function of $t$, it is also interesting to plot $\theta$ vs $\omega$. This is a phase-space plot. Such a plot can be used to distinguish chaotic and periodic solutions. For the non-chaotic case, the motion of the system is characterized by an initial transient that dies out, leaving a regular, steady-state motion with a period given by an integer multiple of $T_d = 2\pi/\omega_d$, the period of the driving term. Consider examining the state of the pendulum at regular intervals $t = nT_d$, defined by the pair of values $[\theta(nT_d),\omega(nT_d)]$. The series of such points for $n = 0,1,2,\ldots{}$ is called a Poincaré section. Steady-state motion with period $T_d$ corresponds to a constant sequence for the Poincaré section. Steady-state motion with period $2T_d$ results in a Poincaré section that alternates between two points in phase space. For the parameter set given here and for $g < 0.97$, the period is just $T_d$. For slightly larger $g$, e.g. $g = 1.07$ the period is $2T_d$. This phenomenon of period doubling is characteristic of a wide class of systems on the approach to chaos. Further increases in $g$ lead to further doubling, until eventually the threshold for chaos is crossed. At this point the Poincaré section has no evident limiting behavior.

Fourier analysis provides another technique for analyzing the system. Here we simply feed the time series $([t,\theta(t)]$ into a Fourier analysis code and plot the resulting power spectrum. Nonchaotic steady state motion shows cleanly defined peaks at the fundamental frequency $1/T_d$ and its harmonics. Chaotic motion shows a rather noisy Fourier spectrum, but with interesting regularities. We do not have time in this course to investigate the intricacies of chaotic behavior, but you are welcome to explore on your own.